Let $\Omega \subset \mathbb{R^n}$ for $n \ge 3$ be a bounded domain with smooth boundary and let $\alpha \in (0,1)$ and $\lambda \in \mathbb{R}$ such that for all $f \in C^{0, \alpha}(\bar \Omega)$, the boundary value problem $$-\Delta u+\lambda u=h$$ on $\Omega$ , $$u=0$$ on $ \partial\Omega$
has unique solution $u$ in $C^{2,\alpha}(\bar \Omega)$. Then show that there exists $\epsilon, \eta$ such that $\forall h \in C^{0,\alpha}(\bar \Omega)$ such that $||h||_{C^{0,\alpha}(\bar \Omega)}< \epsilon$ and $||u||_{C^{0,\alpha}(\bar \Omega)}< \eta$ there exists a unique $u$ in $C^{2,\alpha}(\bar \Omega)$ $$ -\Delta u + \lambda u -u^{\epsilon}=h $$ on $\Omega$
$$u=0 $$ on $\partial\Omega $
I'm not really sure what to do here although I sense the local inverse theorem is the way to go. Thoughts?
Here is a picture about the theorem:
